Vectors in mechanics

A particle P of mass 3 kg is moving on a smooth horizontal surface under the action of two constant horizontal forces (5i + 3j)N and (ai + 3bj)N. The acceleration of P is (2i - 3j) m s^-2.

Find the value of a and the value of b.

At time t = 0 seconds the velocity of P is u m s^-1 and at time t = 5 seconds the velocity of P is (7i - 6j) m s^-1.

Find, in terms of i and j, an expression for u.

A particle P is moving with constant acceleration (-5i + 2j)m s^-2. At time t = 0 seconds, P is at the origin and has velocity (i + 3j)m s^-1.

Find, in terms of i and j, the displacement of P at time t = 2 seconds.

Determine the speed of P at time t = 2 seconds.

In this question the unit vectors i and j are in the directions east and north respectively.

A particle P is moving on a smooth horizontal surface under the action of a single force F N. At time t seconds, where t ⩾ 0, the velocity v ms^-1 of P, relative to a fixed origin O, is given by

v = (1 - 2t)i + (2t² + t - 13)j.

Show that P is never stationary.

Find, in terms of i and j, the acceleration of P at time t.

The mass of P is 0.5 kg.

Determine the magnitude of F when P is moving in the direction of the vector - 2 i + j. Give

your answer correct to 3 significant figures.

When t = 1, P is at the point with position vector .

Determine the bearing of P from O at time t = 1.5.

In this question the unit vectors i and j are in the directions east and north respectively.

At time t seconds, where t ⩾ 0, a particle P of mass 2 kg is moving on a smooth horizontal surface under the action of a constant horizontal force (- 8 i - 54 j)N and a variable horizontal force (4ti + 6(2t - 1)²j)N.

Determine the value of t when the forces acting on P are in equilibrium.

It is given that P is at rest when t = 0.

Determine the speed of P at the instant when P is moving due north.

Determine the distance between the positions of P when t = 0 and t = 3.

A particle P moves with constant acceleration (-4i + 2j)ms^-2. At time t = 0 seconds, P is moving with velocity (7i + 6j)ms^-1.

Determine the speed of P when t = 3.

Determine the change in displacement of P between t = 0 and t = 3.

In this question the unit vectors i and j are in the directions east and north respectively.

A particle R of mass 2 kg is moving on a smooth horizontal surface under the action of a single horizontal force F N. At time t seconds, the velocity v ms^-1 of R, relative to a fixed origin O, is given by v = (pt² - 3t)i + (8t + q)j, where p and q are constants and p < 0.

Given that when t = 0.5 the magnitude of F is 20, find the value of p.

When t = 0, R is at the point with position vector (2 i - 3 j) m.

Find, in terms of q, an expression for the displacement vector of R at time t.

When t = 1, R is at a point on the line L, where L passes through O and the point with position vector 2 i - 8 j.

Find the value of q.

Find the acceleration vector of the particle.

Find the position vector of the particle after 10 seconds.